Theorem cnvepresdmqs

Description: The domain quotient predicate for the restricted converse epsilon relation is equivalent to the negated elementhood of the empty set in the restriction. (Contributed by Peter Mazsa, 14-Aug-2021)

	Ref	Expression
Assertion	cnvepresdmqs	$$⊢\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)DomainQsA\leftrightarrow \neg \varnothing \in A$$

Proof

Step	Hyp	Ref	Expression
1		df-dmqs	$$⊢\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)DomainQsA\leftrightarrow \mathrm{dom}\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)/\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)=A$$
2		n0el3	$$⊢\neg \varnothing \in A\leftrightarrow \mathrm{dom}\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)/\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)=A$$
3	1 2	bitr4i	$$⊢\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)DomainQsA\leftrightarrow \neg \varnothing \in A$$